To check for actual support, run mythfrontend and go into Setup -> Audio. If the E-AC3/DTS-HD flag is enabled, it is likely that your audio card supports it.
Remember you must use an HDMI connection. Optical or coaxial SPDIF will *not* work.

TrueHD and DTS-HD MA support

For your audio card to support loss-less HD audio passthrough (TrueHD and DTS-HD MA), your audio card has to support HBR.
You can check DTS-HD/TrueHD support on Linux by checking for "HBR" text in "Pincap" line in /proc/asound/cardX/codec#Y. This doesn't guarantee in any way that your audio card will support HBR ; but if you don't have it, it's very unlikely that it does.

Chips known to support HBR:

ION2

nVidia GT4xx

nVidia GT5xx

Chips known not to support HBR:

ION first generation

nVidia 9400M IGP

nVidia GT2xx (*)

nVidia 8xxx, 9xxx

(*) Though not officially supported, HBR works to some extent with nVidia GT2xx ; your mileage may vary. It is highly dependent of the combination between audio processor/amplifier and video card.

You'll need a recent linux kernel (>= 2.6.36) that contains the required ALSA drivers (>= 1.0.24. Version can be found in /proc/asound/version).

Alsa library 1.0.23 is the required minimum. However if you had to mess with the device name to get basic audio working, it is strongly recommended you get a new version as otherwise you will likely have to mess with alsa configuration file.

Checking your TV / Amplifier / Audio Processor capabilities

With ALSA 1.0.25 or later, Linux kernel 3.3 or later

Provided your hdmi device (TV or amp) was powered on before your PC started, Myth can display the audio capabilities of the device.

Go into Setup -> Audio. Select one of the ALSA:hdmi audio output device. In the description section, it will show the name of the connected device as well as the list of all the digital codec supported being one of the following:

undefined

LPCM

AC3

MPEG1

MP3

MPEG2

AAC-LC

DTS

ATRAC

DSD (One Bit Audio aka SACD)

E-AC3

DTS-HD

TrueHD

DST

WMAPro

HE-AAC

HE-AACv2

MPEG Surround

As of Myth 0.25, only LPCM, AC3, DTS, DTS-HD and TrueHD are supported, with the caveats mentioned in the earlier section.

When connected to a Primare SP32 amplifier:

When connected to a Yamaha SP-4000 soundbar (connected to a Sony TV):

If the description contains the word "guessing: ..."; then myth hasn't been able to retrieve information from the TV or amp. Digital bitstreaming isn't guaranteed to work under those circumstances

For earlier audio drivers

For HDA Intel audio device (used pretty much for all HDMI devices), recent version of linux kernel or alsa, it is usually possible to retrieve what your audio processing capabilities are:

The file is located in /proc/asound/cardX/eld#Y.0 (where X is the card number, and Y is the device number starting from 0. So if aplay -l shows Card 0, device 3 as being the first device listed
you need to check /proc/asound/card0/eld#0.0).

Summary of working / non-working known solution

HBR: is a flag that would appear in the eld file above. Put - if eld doesn't exist, or HBR isn't mentioed

LPCM: If eld exists, enter the number of channels in the LPCM section. If the eld file doesn't exist, enter the maximum number of channels as reported by alsa. You can get this information running mythfrontend with -v audio.